Todd Trimble monomorphisms in the category of groups

Proof

Let $i: H \to G$ be monic, and let $\pi: G \to G/H$ be the canonical surjective function $g \mapsto g H$ . Let $A$ be the free abelian group on $G/H$ with $j: G/H \to A$ the canonical injection, and let $A^G$ denote the set of functions $f: G \to A$, with the pointwise abelian group structure inherited from $A$. This carries a $G$-module structure defined by

For any $f \in A^G$, the function $d_f: G \to A^G$ defined by $d_f(g) = g f - f$ defines a derivation. Passing to the wreath product $A^G \rtimes G$, we have two homomorphisms $\phi, \psi: G \rightrightarrows A^G \rtimes G$ defined by $\phi(g) := (d_{j \pi}(g), g)$ and $\psi(g) := (0, g)$. I claim that $i: H \to G$ is the equalizer of the pair $\phi, \psi$. For,

(All we needed was some injection $j: G/H \to A$ into an abelian group; I chose the canonical one.)

Proof

This follows because an epic equalizer is an equalizer of two maps that are the same, hence an isomorphism.

Proof

Since every morphism $f: G \to H$ factors as a regular epi $p: G \to G/\ker(f)$ followed by a mono $i$, having $f$ epic implies $i$ is a epic mono. Epic monos $i$ being isomorphisms, $f$ is then forced to be regular epic as well.